MEMO 2009 Problem 9

Let $ABCD$ be a parallelogram with $\angle BAD = 60$ and denote by $E$ the intersection of its diagonals. The circumcircle of triangle $ACD$ meets the line $BA$ at $K \ne A$ , the line $BD$ at $P \ne D$ and the line $BC$ at $L\ne C$ . The line $EP$ intersects the circumcircle of triangle $CEL$ at points $E$ and $M$ . Prove that triangles $KLM$ and $CAP$ are congruent.